You can run a second EFA using the estimated correlation matrix as you say. This would have two drawbacks, however. First, you can only use the ULS estimator since you don't then have the raw data to compute the weight matrix of WLSMV. Second, the estimated correlation matrix does not have the second-order structure imposed on it. You can instead do an "EFA within a CFA framework" and do the second-order modeling directly. You may also want to consult the old psychometric literature on "Schmid-Leiman orthogonalization".